We propose an experiment to search for QCD axion and axion-like-particle (ALP) dark matter. Nuclei that are interacting with the background axion dark matter acquire time-varying CP-odd nuclear moments such as an electric dipole moment. In analogy with nuclear magnetic resonance, these moments cause precession of nuclear spins in a material sample in the presence of a background electric field. This precession can be detected through high-precision magnetometry. With current techniques, this experiment has sensitivity to axion masses m_a <~ 10^(-9) eV, corresponding to theoretically well-motivated axion decay constants f_a >~ 10^16 GeV. With improved magnetometry, this experiment could ultimately cover the entire range of masses m_a <~ 10^(-6) eV, just beyond the region accessible to current axion searches.